Hopf Ideals, Integrality, and Automorphisms of Quantum Groups at Roots of 1

TL;DR

This paper classifies skew-commutative subalgebras in quantum groups at roots of unity, explores associated Hopf ideals, and constructs restricted quantum groups, extending prior results to even orders and non-simply laced types.

Contribution

It provides a comprehensive classification of subalgebras and Hopf ideals in quantum groups at roots of unity, including new cases and explicit computational descriptions.

Findings

Classified centrality and commutativity of skew-polynomial algebras depending on Lie type and root order.

Constructed and analyzed Hopf ideals related to Weyl group elements and Bruhat subgroups.

Extended results to even roots of unity and non-simply laced Lie types, with explicit algebraic descriptions.

Abstract

We consider skew-commutative subalgebras in Drinfeld-Jimbo quantum groups at a root of unity $\zeta$ generated by primitive power elements. We classify the centrality and commutativity of these skew-polynomial algebras depending on the Lie type and the order of $\zeta$ modulo 8. We describe Hopf ideals in the quantum group induced from ideals in these subalgebras, including the non-commutative cases. Among these, we construct and analyze a family of Hopf ideals that depend on the choice of an element in the Weyl group. We show that they arise naturally both in the construction of (partial) $R$-matrices and as vanishing ideals of Bruhat subgroups. Specialization to the maximal element yields a rigorous construction of restricted quantum groups as pre-triangular Hopf algebras, independent of any choices. Our treatment also includes even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings. Consequently, we extend some results of De Concini-Kac-Procesi, whose work focuses on odd orders of $\zeta$, which forces the subalgebra to be strictly central, and complex ground fields. This includes the identification of the subalgebras for Lie types $\mathsf{A}_n$ and $\mathsf{B}_2$ with the coordinate rings of associated algebraic groups in the commutative cases, even if $\zeta$ has even order. Our descriptions are computationally explicit and do not utilize Poisson structures. As technical preparations, we discuss PBW bases over minimal rings, dependencies on choices of convex orderings, as well as various new constructions of, and relations among, automorphisms on quantum groups. The latter include formulae for the Garside element in the Lustzig-Artin group action and the family of Che-transformations.